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Efficiency of several voting systems with an impartial culture model and 25 voters[1]

Social utility efficiency (SUE)[1] or voter satisfaction efficiency (VSE)[2] is a metric for comparing voting methods which compares them based on the average well-being of voters.

It is also known as utilitarian efficiency[3][4] or voter satisfaction index (VSI).[5][6]

Definition

Social utility efficiency is defined as the ratio between the social utility of the candidate who is actually elected by a given voting method and that of the candidate who would maximize social utility, where is the expected value over many iterations of the sum of all voter utilities for a given candidate:[7]

A voting method with 100% efficiency would always pick the candidate that maximizes voter utility. A method that chooses a winner randomly would have efficiency of 0%, and a (pathological) method that did worse than a random pick would have less than 0% efficiency.

SUE is not only affected by the voting method, but is a function of the number of voters, number of candidates, and of any strategies used by the voters.[1]

History

The concept was originally introduced as a system’s “effectiveness” by Robert J. Weber in 1977, defined as:[8]

Where is the expected social utility of the given candidate, is the number of voters, and is the number of candidates. He used a random society (impartial culture) model to analytically calculate the effectiveness of FPTP, two Approval variants, and Borda, as the number of voters approaches infinity.

It was given the name “social utility efficiency” and extended to the more realistic spatial model of voting by Samuel Merrill III[1] in the 1980s, calculated statistically from random samples, with 25–201 voters and 2–10 candidates. This analysis included FPTP, Runoff, IRV, Coombs, Approval, Black, and Borda (in increasing order of efficiency).[9][10] (Merrill’s model normalizes individual voter utility before finding the utility winner, while Weber’s does not, so that Merrill considers all 2-candidate voting systems to have an SUE of 100%, decreasing with more candidates, while Weber considers them to have an effectiveness of = 81.6%, with some systems increasing with more candidates.)

In 2017, Jameson Quinn studied SUE under the name “voter satisfaction efficiency”,[11] using more complex and arguably more realistic parameters, examining a wider variety of scenarios and using a hierarchical cluster model of voter behavior. He found the best performers to be ranked pairs, STAR voting, and score voting, depending on the scenario tested.[11][12][2][13]

A similar metric, referred to as “Bayesian regret“,[14][15][16] measures the same property, but inverted.[17][18][19] They are related by the formula:[11]

where “random winner” refers to the hypothetical election method of choosing a candidate at random regardless of the opinions of the electorate (not the random ballot voting method, which is weighted towards candidates who receive more votes).

While the term “utilitarian efficiency” has been used as a synonym for SUE, it has also been used to mean the probability of electing the utilitarian winner (analogous to Condorcet efficiency).[20][21][22]

In 2023, Robinette introduced a variant of social utility efficiency that compares voting rules under strategic candidate repositioning, measuring the utility of the repositioned winning candidate against the maximum utility achievable from the candidates’ initial positions.[23]

In 2025, Holliday and Pacuit introduced expected social utility performance (ESUP), a variant of SUE that averages the per‑election normalized utility rather than the ratio of expected utilities across all elections. They conducted a large‑scale simulation across 36 voting methods, using a spatial model with 15 parameters and nearly one million CPU hours, confirming that Condorcet and Borda methods achieve the highest expected social utility, while IRV and Plurality perform significantly worse.[24]

See also

References

  1. ^ a b c d Merrill, Samuel (1984). “A Comparison of Efficiency of Multicandidate Electoral Systems”. American Journal of Political Science. 28 (1): 23–48. doi:10.2307/2110786. ISSN 0092-5853. JSTOR 2110786. social utility efficiency is defined as the ratio between the expected social utilities of the candidate selected by the system and the candidate maximizing social utility
  2. ^ a b Wolk, Sara; Quinn, Jameson; Ogren, Marcus (2023-03-20). “STAR Voting, equality of voice, and voter satisfaction: considerations for voting method reform”. Constitutional Political Economy. 34 (3): 310–334. doi:10.1007/s10602-022-09389-3. ISSN 1043-4062. S2CID 257653868.
  3. ^ Mueller, Dennis C. (2003). Public choice III. Cambridge: Cambridge University Press. ISBN 0-511-06504-3. OCLC 191952945. An alternative normative criterion to that of Condorcet efficiency for a voting procedure is that it should maximize a utilitarian welfare function of … cardinal interpersonally comparable utility indexes for each voter
  4. ^ Duddy, Conal (2017). “Geometry of run-off elections”. Public Choice. 173 (3–4): 267–288. doi:10.1007/s11127-017-0476-2. ISSN 0048-5829. S2CID 254935333. Social utility efficiency, also known as utilitarian efficiency, relates the social utility (sum of voter utilities) of the candidate elected by a rule to that of the utility-maximizing candidate
  5. ^ Shentrup, Clay (2007-07-07). “Voter Satisfaction Index”. Center for Range Voting. Retrieved 2019-07-24. Voter satisfaction index, or “VSI” for short (also called “social utility efficiency” … a lower number is actually better, and this can confuse people who are new to the concept. … the utility units have an arbitrary magnitude, making it difficult to compare Bayesian regret figures
  6. ^ Huang, John (January 11, 2020). “Alternative Voting Methods — How well do they perform in the best case?”. Americans for Representation. Retrieved 2021-01-31. For this blog post, I’m going to stick with something called “Voter Satisfaction Index”.
  7. ^ Merrill, Samuel (2014-07-14). Making Multicandidate Elections More Democratic. Princeton University Press. ISBN 9781400859504. If the ratings are interpreted as Von Neumann-Morgenstern utilities … I define the social utility of a candidate as the sum of all voter utilities for that candidate.
  8. ^ Weber, Robert J. (September 1978). “Comparison of Public Choice Systems”. Cowles Foundation Discussion Papers. Cowles Foundation for Research in Economics: 16, 38, 62. No. 498.
  9. ^ Plassmann, Florenz; Tideman, T. Nicolaus (2014-01-01). “How frequently do different voting rules encounter voting paradoxes in three-candidate elections?”. Social Choice and Welfare. 42 (1): 31–75. doi:10.1007/s00355-013-0720-8. ISSN 0176-1714. To our knowledge, Merrill (1984) provided the only previous empirical assessment of the Black rule—a calculation of the social-utility efficiency of this rule.
  10. ^ Kondratev, Aleksei Y.; Nesterov, Alexander S. (2020-04-01). “Measuring majority power and veto power of voting rules”. Public Choice. 183 (1–2): 187–210. arXiv:1811.06739. doi:10.1007/s11127-019-00697-1. ISSN 0048-5829. Statistically, Black’s rule is much less manipulable than the Borda rule … however, the Borda rule provides a slightly higher social utility efficiency (Merrill,1984).
  11. ^ a b c “Voter Satisfaction Efficiency Simulator”. Jameson Quinn. Retrieved 2021-02-03.
  12. ^ Frohnmayer, Mark. “The Election Science Behind the Reform Movement”. Equal Vote. Retrieved 28 December 2020.
  13. ^ Hamlin, Aaron; Hua, Whitney (2023-09-01). “The case for approval voting”. Constitutional Political Economy. 34 (3): 335–345. doi:10.1007/s10602-022-09381-x. ISSN 1043-4062. Recent research using computer simulations under a Monte Carlo method demonstrates that approval voting also produces winners that reliably maximize voter satisfaction
  14. ^ Smith, Warren D. (2006). “Bayesian Regret for dummies”. RangeVoting.org. Retrieved 2021-01-31.
  15. ^ McKenna, Phil (2008-04-01). “Vote of no confidence”. New Scientist. 198 (2651): 30–33. doi:10.1016/S0262-4079(08)60914-8. Smith bases these claims on a series of computer simulations … he measured “Bayesian regret”, a parameter that attempts to quantify how unhappy groups of people are following a poor outcome. The better the voting system, the less Bayesian regret it causes.
  16. ^ Poundstone, William (2008). Gaming the vote: why elections aren’t fair (and what we can do about it) (1st ed.). New York: Hill and Wang. ISBN 978-0-8090-4893-9. Bayesian regret” is a statistical term that Smith defines as “expected avoidable human unhappiness.
  17. ^ Hansen, Jeremy A (2014). “Comparing Approval At-Large to Plurality At-Large in Multi-Member Districts”. Conference: Fifth International Workshop on Computational Social Choice. Social-utility efficiency … Smith referred to a similar formulation as Bayesian regret
  18. ^ “Range voting with mixtures of honest and strategic voters”. RangeVoting.org. Retrieved 2019-07-26. SociallyBest 0 … RandomWinner 1 … SociallyWorst 2.0024
  19. ^ Smith, Warren D. (1 September 2023). “The case for score voting”. Constitutional Political Economy. 34 (3). Springer Nature: 297–309. doi:10.1007/s10602-023-09403-2. Retrieved 30 November 2024.
  20. ^ Green-Armytage, James; Tideman, T. Nicolaus; Cosman, Rafael (2016-01-01). “Statistical evaluation of voting rules”. Social Choice and Welfare. 46 (1): 183–212. doi:10.1007/s00355-015-0909-0. ISSN 0176-1714. Two measures of efficiency are predominant here: the share of trials in which the candidate that maximizes the sum of utilities is chosen, and the average share of available welfare that is produced by the chosen candidate.
  21. ^ Lehtinen, Aki (2007-09-10). “The Borda rule is also intended for dishonest men”. Public Choice. 133 (1–2): 73–90. doi:10.1007/s11127-007-9178-5. ISSN 0048-5829. Utilitarian efficiency is defined as the percentage of simulated voting games in which the candidate with the highest utility sum (the utilitarian winner) is selected.
  22. ^ Lehtinen, Aki (2008-09-01). “The welfare consequences of strategic behaviour under approval and plurality voting”. European Journal of Political Economy. 24 (3): 688–704. doi:10.1016/j.ejpoleco.2008.03.002. Utilitarian efficiency is defined as the percentage of simulated voting games in which the candidate that maximises the sum of voters’ utilities (the utilitarian winner) is selected
  23. ^ Robinette, Robbie (2023-09-01). “Implications of strategic position choices by candidates”. Constitutional Political Economy. 34 (3): 445–457. doi:10.1007/s10602-022-09378-6. ISSN 1043-4062. I propose a refinement to the social utility efficiency metric to account for the different utility of the candidate’s chosen positions
  24. ^ Holliday, Wesley H.; Pacuit, Eric (2025), The Social Utility of Voting Revisited, doi:10.2139/ssrn.5073085, retrieved 2026-05-07